In x-ray computed tomography, the weakening intensity of the radiation along different projection beams through an object being examined is measured and used to deduce the density distribution in the object. Analytical reconstruction methods to reconstruct the object from the projection data are based on an inversion of the x-ray transform which the model of the x-ray computed tomography represents.
A reconstruction method described in the technical article “Mathematical framework of cone-beam reconstruction via the first derivative of the radon transform” by P. Grangeat in G. T. Herman, A. K. Louis and F. Natterer, Editors, Lecture Notes in Mathematics, Volume 1497, pp. 66 to 97, Springer 1991 postulates the existence of a special trajectory designated as the Tuy curve. The inversion formula underlying the reconstruction method is based on a relationship between the known 3-dimensional radon transform and the known 3-dimensional x-ray transform. In using this reconstruction method, a derivative has to be formed for a discontinuous function known as a Crofton function, which generally results in numerical instabilities in the reconstruction method, due to which the reliability and accuracy of the reconstruction method is unsatisfactory.
The technical article “A general scheme for constructing inversion algorithms for cone beam ct.” by A. Katsevich, International Journal of Mathematics and Mathematical Sciences, 2003 (21): 1305 to 1321, 2003, discloses a reconstruction method based on the relationship between the known 3-dimensional radon transform and the known 3-dimensional x-ray transform described in the technical article by P. Grangeat, but which uses a weighting function to avoid the derivative of the Crofton function. While the reconstruction method is numerically more stable, the establishment of the weighting function is laborious, however.